In a communication network, there are two types of packets. Header-type packets contain control information such as source and destination addresses and checksum bytes etc. whereas user-generated information is in data packets. Assume that a message consists of 6 header packets followed by 94 data packets. However a faulty router re-orders all of the packets randomly. How many random ordering of 100 packets is possible? 100!94!94100What is the probability that the first header-type packet to be received is the 10th packet to arrive? 0.04220.03690.01890.0256
Question
In a communication network, there are two types of packets. Header-type packets contain control information such as source and destination addresses and checksum bytes etc. whereas user-generated information is in data packets. Assume that a message consists of 6 header packets followed by 94 data packets. However a faulty router re-orders all of the packets randomly. How many random ordering of 100 packets is possible? 100!94!94100What is the probability that the first header-type packet to be received is the 10th packet to arrive? 0.04220.03690.01890.0256
Solution
- How many random ordering of 100 packets is possible?
This is a permutation problem with repetition. The formula for permutations with repetition is n! / (n1! * n2! * ... * nk!), where n is the total number of items, and n1, n2, ..., nk are the numbers of each type of item.
So, the number of random orderings of 100 packets is 100! / (6! * 94!) = 1192052400.
- What is the probability that the first header-type packet to be received is the 10th packet to arrive?
This is a hypergeometric distribution problem. The formula for the hypergeometric distribution is P(X = k) = [C(K, k) * C(N-K, n-k)] / C(N, n), where N is the total number of items, K is the total number of success states in the population, n is the number of draws, and k is the number of observed successes.
So, the probability that the first header-type packet to be received is the 10th packet to arrive is P(X = 1) = [C(6, 1) * C(94, 9)] / C(100, 10) = (6 * 1.73 * 10^11) / 2.63 * 10^13 = 0.0393 or 3.93%. The closest answer is 0.0422.
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